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  "body": "\n## Definition\n\nLet $X$ and $Y$ be sets, let $k \\ge 1$, and let $D \\subseteq X^k$.\nLet $f \\colon X \\to Y$ be an unknown [[function|0EPAT51DXQ]].\nLet $E_1(f)$ and $E_2(f)$ be expressions built from $f$, from given functions and from variables $x_1, \\dots, x_k$ by finitely many substitutions and compositions.\nA *functional equation* in $f$ is a condition\n$$E_1(f)(x_1, \\dots, x_k) = E_2(f)(x_1, \\dots, x_k) \\quad \\text{for all } (x_1, \\dots, x_k) \\in D.$$\nIn particular the expressions contain no derivatives, integrals or limits of $f$.\nA *solution* is a function $f \\colon X \\to Y$ that satisfies the condition.\nSystems of such conditions and several unknown functions are allowed in the same way.\n\n## Remarks\n\nExample: Jensen's equation $f\\!\\left(\\frac{x + y}{2}\\right) = \\frac{f(x) + f(y)}{2}$ on $D = \\R^2$ is a functional equation with $X = Y = \\R$.\nThe function $f(x) = 3x + 1$ solves it, since both sides equal $\\frac{3(x + y)}{2} + 1$.\n\nNon-example: $f'(x) = f(x)$ for all $x \\in \\R$ is not a functional equation in this sense, because it involves the derivative of $f$; it is a differential equation.\n\nDifference equations such as $f(n + 2) = f(n + 1) + f(n)$ on $\\Z$ are functional equations in one variable.\nIterative equations, such as $f(f(x)) = g(x)$ for a given $g$, are another important class.\n",
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